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Statistical Methods for Economic Time Series

When estimating relationships, making
forecasts and testing hypotheses from economic theory,
researchers frequently use data in the form of time series
– chronological sequences of observations – to
study macroeconomic variables. Consumption in an economy
may thus depend on total labor income and wealth, real
interest rates, the age distribution of the population,
etc. The simplest conceivable textbook example of such a
relationship is a static, linear expression with only two
variables:

According to this equation, the
variable gt (for
instance, consumption in quarter t) depends on the variable
xt (for instance, income during the
same period). The last, random-error, term
et denotes the variation in gt which cannot be explained by
the model. By means of time series for the variables
gt and
xt, the parameters a and b can be
estimated using statistical methods (known as regression
analysis). Valid conclusions presuppose that the methods
are well adapted to the specific properties of the time
series. This year’s laureates have developed methods
that capture two key properties of many economic time
series: nonstationarity and time-varying
volatility.

Nonstationarity,
Common Trends and Cointegration

Many macroeconomic time series are nonstationary: a
variable, such as GDP, thus follows a long-run trend, where
temporary disturbances affect its long-term level. In
contrast to stationary time series, nonstationary series do
not exhibit any clear-cut tendency to return to a constant
value or a given trend. Figure 1 shows two
examples of such time series. The jagged curve, with large
short-run variations, represents the exchange rate between
the Japanese yen and the U.S. dollar for each month since
1970. The smoother curve shows the consumer price level in
Japan in relation to that in the U.S. during the same
period.

Figure 1: Logarithm of the Japanese
yen/U.S. dollar exchange-rate index and the logarithm
of the quotient between the consumer price index for
Japan and the consumer price index for the U.S.;
monthly observations, January 1970–May 2003.

Statistical
Pitfalls

For a long time, despite the fact that macroeconomic time
series are often nonstationary, researchers only had access
to standard methods developed for stationary data. In 1974,
Clive Granger (and his colleague Paul
Newbold) demonstrated that estimates of relationships
between nonstationary variables could yield nonsensical
results by erroneously indicating significant relationships
between wholly unrelated variables. (In the above equation,
the problem arises if the random error
et is nonstationary. A standard test
may then indicate that b is
different from 0, even though the true value is 0.)

Statistical pitfalls can also give
rise to misleading results in cases where a relationship
does in fact exist. In particular, it may be difficult to
distinguish between temporary and permanent relationships
among nonstationary time series. For example, economic
theory postulates that, in the long run, a stronger
exchange rate should be associated with relatively slower
price increases, because prices expressed in a common
currency cannot deviate too much from one another. Such a
tendency is also revealed in Figure 1, where the
yen became stronger against the dollar over the period,
while the price level in the U.S. rose in relation to the
Japanese price level. In the short run, however,
expectations and capital movements have such a pervasive
effect on the exchange rate that standard methods may be
inadequate for precise estimation of the long-run
relationship.

A common approach to dealing with the
problem of nonstationary data had been to specify
statistical models as relationships between differences,
i.e., rates of increase. Instead of using the exchange rate
and the relative price level, one would estimate the
relationship between currency depreciation and relative
inflation. If the rates of increase are indeed stationary,
traditional methods provide valid results. But even if a
statistical model based solely on difference terms can
capture the short-run dynamics in a process, it has less to
say about the long-run covariation of the variables. This
is unfortunate because economic theory is often formulated
in terms of levels and not differences.

Owing to the properties of
nonstationary data, it therefore became a challenge to find
methods which could trace the potential long-run
relationships concealed by the noise of short-run
fluctuations. The work of Clive Granger has generated such
a methodology for statistical analysis.

Granger's
Contribution

In research published during the 1980s, Granger developed
concepts and analytical methods that combine short-run and
long-run perspectives. The key to these methods, and to
valid statistical inference, is his discovery that a
specific combination of two (or more) nonstationary series
may be stationary. Economic theory often makes exactly such
predictions: if there is an equilibrium relationship
between two economic variables, they may deviate from the
equilibrium in the short run, but will adjust towards the
equilibrium in the longer run. For example, conventional
theory predicts a long-term equilibrium exchange rate,
where price levels expressed in a common currency are on
parity with each other. Granger minted the term
cointegration for a stationary combination of
nonstationary variables.

Granger also demonstrated that the
joint dynamics among cointegrated variables may be
expressed in a so-called error-correction model. Such a
model is not only statistically sound, but can also be
given a meaningful economic interpretation. For example,
the dynamics in exchange rates and prices are driven by two
simultaneous forces: a tendency to smooth out deviations
from the long-run equilibrium exchange rate, and short-run
fluctuations around the adjustment path towards this
long-run equilibrium.

The concept of cointegration would
not have become useful in practice without powerful
statistical methods for estimation and testing of
hypotheses. Clive Granger and Robert Engle introduced such
methods in a remarkably influential article published in
1987. Here, they present a test of the hypothesis that a
number of nonstationary variables are not cointegrated, as
well as a two-step method for estimating the
error-correction model. Improved methods, which have now
become standard, were later developed by Søren
Johansen.

In subsequent work and in
collaboration with other researchers, Granger has extended
cointegration analysis in several respects, including the
ability to handle series with seasonal patterns (seasonal
cointegration) and series where adjustment towards
equilibrium does not occur until the deviation exceeds a
critical value (threshold cointegration).

Applications

Clive Granger’s work has transformed the way
economists deal with time-series data. Today, tests of
stationarity and cointegration are carried out routinely as
a stepping-stone to the specification of dynamic
econometric models. Cointegration analysis has turned out
to be particularly valuable in systems where short-term
dynamics are affected by large random disturbances, while
long-term variations are simultaneously constrained by
economic equilibrium relationships. An example is the
relation between exchange rates and price levels. Other
examples include the relation between consumption and
wealth (which have to be consistent with one another in the
long run, although consumption is much smoother than wealth
in the short run), dividends and stock prices (where stock
prices follow the development of dividends in the long run,
but exhibit substantially larger fluctuations in the short
run) and interest rates of different maturities (where long
and short rates are linked together by expectations
regarding future short rates, even if they move in
different directions in the short run).

Time-Varying Volatility and Arch

Risk evaluation is at the core of
activities on financial markets. Investors assess expected
returns of an asset against its risk. Banks and other
financial institutions would like to ensure that the value
of their assets does not fall below some minimum level that
would expose the bank to insolvency. Such evaluations
cannot be made without measuring the volatility of asset
returns. Robert Engle developed improved
methods for carrying out these kinds of evaluations.

Figure 2 shows the returns
on an investment in the NYSE stock index (the Standard
& Poor 500) for all stock-market days between May 1995
and April 2003. The returns averaged 5.3 percent per year.
At the same time there were days, when the fluctuations in
prices were greater (plus or minus) than 5 percent. The
standard deviation* in daily returns
measured over the entire period was 1.2 percent. Closer
inspection reveals, however, that the volatility varies
over time: large changes (upwards or downwards) are often
followed by further large fluctuations, and small changes
tend to be followed by small fluctuations. This is clearly
illustrated in Figure 3, which shows how the
standard deviation, measured over the last four weeks,
moved over time. Evidently, the standard deviation varied
considerably, from approximately 0.5 percent during calm
periods to nearly 3 percent during more turbulent episodes.
Many financial time series are characterized by similar
time variation in volatility.

Figure 3: Standard deviation for
percentage daily returns on an investment in the
Standard & Poor 500 stock index, May 16,
1995–April 29, 2003, computed from data for the
four preceding weeks.

Engle’s
Contribution

Figure 3 shows backward-looking calculations of
time-varying volatility. But investors and financial
institutions need forward-looking evaluations –
forecasts – of volatility during the next day, week
and year. In an outstanding article in 1982, Robert Engle
formulated a model which allows such evaluations.

Statistical models of asset returns
can only explain a fraction of the variation from one day
to the next. Most of the volatility is thus embedded in the
random error term (et in the
introductory equation) – or, in other words, in the
model’s forecasting error. In standard statistical
models, the expected variance of the random error is
assumed to be constant over time. Obviously, this is far
from capturing the large variations in asset returns
depicted in Figure 3.

Engle assumed instead that the
variance of the random error in a certain statistical
model, in a certain time period, systematically depends on
previously realized random errors, so that large (small)
errors tend to be followed by large (small) errors. In
technical terms, the random variable displays
autoregressive conditional heteroskedasticity. His
approach has therefore become acronymized ARCH. In our
example, the model now contains not only a forecasting
equation for asset returns, but also a number of parameters
showing how the variance of the random error in this
equation depends on forecasting errors in earlier periods.
Engle demonstrated how ARCH models could be estimated and
introduced a practical test for the hypothesis that the
conditional variance of the random error is constant.

In subsequent work and in
collaboration with students and colleagues, Engle developed
this concept in several different directions. The
best-known extension is the generalized ARCH model (GARCH)
developed by Tim Bollerslev in 1986. Here, the variance of
the random error in a certain period depends not only on
previous errors, but also on the variance itself in earlier
periods. This development has turned out to be very useful;
GARCH is the model most often applied today.

Applications

In his first article on ARCH, Engle used his model of
time-varying volatility to study inflation. Not before
long, however, it became clear that the most important
applications were to be found in the financial sector,
where activities aim at handling and pricing different
types of risk. Price-setting models thus represent the
relation between prices of securities and volatility: the
expected returns on specific shares depend on the
covariance between the return on the share and the market
portfolio (according to the CAPM developed by Sharpe,
Economics Laureate in 1990), option prices depend on the
variance in the return on the underlying asset (according
to the Black-Scholes formula, awarded the Economics Prize
in 1997 to Merton and Scholes), etc.

In joint work with other researchers,
Engle has captured these relationships by developing models
(GARCH-M) where expected returns depend on time-varying
variances and covariances, thereby becoming time-varying
themselves.

What are the practical implications
of time-varying volatility? If a GARCH model is applied to
the stock returns in Figure 2, conditional
volatility, expressed as a standard deviation, fluctuates
between 0.5 and 3 percent during the period in question. If
an investor has a portfolio corresponding to the Standard
& Poor 500, how much capital would she risk losing the
next day? Given a forecasted standard deviation of 0.5
percent, her loss – with 99 percent probability
– would not exceed 1.2 percent of the value of the
portfolio. If the forecasted standard deviation were 3
percent, the corresponding capital loss would be as high as
6.7 percent. Similar calculations of value at risk
are crucial in modern risk analysis when banks and other
institutions compute the market risk in their securities
portfolios. Since 1996, an international agreement (the
so-called Basle rules) also prescribes the use of value at
risk in the control of banks’ capital requirements.
Through its use in these and other contexts, the ARCH
frame-work is an indispensable tool for risk assessment in
the financial sector.

* The standard
deviation is defined as the square root of the variance,
which gives the average squared deviation from the mean
value of a series. The variance for T observations
of a variable xt with mean value can thus
be computed as